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Geometry Unit 9: Transformations

Transformations move or resize figures. Rigid motions (translate, reflect, rotate) keep size and shape; dilations keep shape but change size.

Covers: Translations, reflections, rotations, dilations, coordinate rules, symmetry, proving shapes on the coordinate plane

Study guide

A transformation moves or resizes a figure. Rigid motions (isometries) keep size and shape, so the image is congruent to the original: translations slide every point the same distance and direction, reflections flip over a line, and rotations turn about a point. Dilations resize by a scale factor k from a center; the image is similar, not congruent, unless k = 1.

Coordinate rules make transformations exact. A translation by (a, b) maps (x, y) → (x + a, y + b). A reflection over the x-axis gives (x, −y), over the y-axis gives (−x, y), and over y = x gives (y, x). Rotations about the origin: 90° counterclockwise gives (−y, x), 180° gives (−x, −y), and 270° counterclockwise gives (y, −x). A dilation by k from the origin gives (kx, ky). Composing transformations means applying one rule and then the next, in order.

A figure has line symmetry if a reflection maps it onto itself and rotational symmetry if a rotation of less than 360° does.

Coordinate proofs use algebra to prove geometric facts: the distance formula shows equal sides, slopes show parallel (equal) or perpendicular (negative reciprocal) sides, and the midpoint formula shows that diagonals bisect each other, which identifies parallelograms, rectangles, rhombuses and squares.

A dilation maps a line to a parallel line (same slope). If the center of dilation lies on the line, the line maps to itself. Otherwise, for a dilation centered at the origin with factor k, the y-intercept is multiplied by k. The scale factor is image length ÷ original length. The center is where the lines through each pair of corresponding points (A and A′, B and B′) meet. For a dilation centered at a point C other than the origin, move each point along ray CP until its distance from C is k times the original.

A regular n-gon maps onto itself under rotations about its center by any multiple of 360°/n. For example, a regular hexagon maps onto itself at 60°, 120°, 180° and so on. A rotation that works for two figures must be a common multiple of both basic angles. Rigid motions preserve distance, angle measure, area, parallelism and collinearity. Dilations preserve angle measure, parallelism and collinearity, but they multiply lengths by k and area by k².

Key ideas

• Translation (x, y) → (x + a, y + b).
• Reflect over the x-axis: (x, −y). Over the y-axis: (−x, y). Over y = x: (y, x).
• Rotate 90° counterclockwise about the origin: (−y, x).
• Dilation by k from the origin: (kx, ky).
• Use slopes and distances to classify quadrilaterals on a grid.

Formulas and rules

Common mistakes

• Mixing up reflection over the x-axis vs y-axis.
• Rotating clockwise when counterclockwise was asked.

Words to know

Reflect (x, y) over the y-axis

(−x, y)

Rotate (x, y) 90° counterclockwise about the origin

(−y, x)

Rotate (x, y) 180° about the origin

(−x, −y)

Rigid motion

A transformation that keeps size and shape: translation, reflection, rotation.

Dilation by factor k from the origin

(kx, ky)

Translation

A slide: (x, y) → (x + a, y + b).

Reflection over the x-axis

(x, y) → (x, −y)

Reflection over y = x

(x, y) → (y, x)

Rotation 270° counterclockwise

(x, y) → (y, −x)

Line symmetry

A reflection maps the figure onto itself.

Rotational symmetry

A rotation under 360° maps the figure onto itself.

Coordinate proof

Using distance, slope and midpoint formulas to prove geometric facts.

Sample practice questions

29 more questions like these, from level 1 to 5, are in the practice set.